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Ok, but the intuition would break then.
If I'm going to fill and drain a bathtub in discrete steps of well defined amounts, only varying the order of operations, then the expectation is that we always end up with the same outcome.
What the analogy proposes is that the baseline is unknown, which is what makes for any order to lead to equally valid yet unequal convergence values. There is then maybe, in the rearrangements, a hidden constant that's inadvertently being changed.
There is then maybe, in the rearrangements, a hidden constant that's inadvertently being changed.
That's the troubling part because it's not true under any number of finite steps.
I'm not familiarized with the series. Maybe if they are rearranged in two infinite series, so that one gives zero right from the start (i.e. for finite steps), the second one might reveal the form of the sthealty constant.
Nope. There are infinitely many ways to divide these kinds of series into one that goes to zero and another that goes to whatever value you want (I'm pretty sure).
I'm sure of that, the thing is that every one of those combinations should reveal the construction of a different constant. That there are infinite ways to do so is the first hint that infinite different constants are obtainable, hence the differing results.
That's partly why I'm so interested in these types of sequences. If they are an inconsistency, I'm not sure how much of the math we rely on would have to be sacrificed.
I think you answered that already. Like for my example, you concluded that in order to keep operating normally, we would have to limit ourselves (depending on the application) to a certain domain. We always have to do that in physics, for instance. Even with something as simple as the equation of the deceleration of a car stopping, you have to limit the domain to the moment it stops, as otherwise the equation predicts it would start accelerating in reverse.
It's perfectly appropriate! You just have to abstract the time away by steps and consider the mass of water added in discrete volumes. Every sum is a new discrete volume of water, and every subtraction is a drained discrete volume of water. As long as the time window for each action is the same, time is irrelevant, and you can just account for the actions in sequential steps, exactly in the form of a series.